Calculation of the 3D interaction geometry for two interlinked chi((2)) processes generating entangled triplets

The geometrical conditions are analytically calculated in which two interlinked second-order nonlinear interactions generating entangled triplets are phase-matched in 3D. The interactions are the spontaneous parametric downconversion of an intense pump and the upconversion of a portion of the downconversion cones by a further pump. The calculation applies to type-I interactions occurring in uniaxial crystals and allows us to forecast positions and frequencies of the entangled triplets experimentally generated. The photon number statistics and the intensity correlations are also measured.